A finite nonabelian simple group does not admit a free action
on a homology sphere, and the only finite simple group which acts on a homology
sphere with at most 0-dimensional fixed point sets ("pseudofree action") is the
alternating group A_5 acting on the 2-sphere. Our first main theorem is the
finiteness result that there are only finitely many finite simple groups which
admit a smooth action on a homology sphere with at most d-dimensional fixed
points sets, for a fixed d. We then go on proving that the finite simple
groups acting on a homology sphere with at most 1-dimensional fixed point sets
are the alternating group A_5 in dimensions 2, 3 and 5, the linear
fractional group PSL_2(7) in dimension 5, and possibly the unitary
group PSU_3(3) in dimension 5 (we conjecture that it does not admit any
action on a homology 5-sphere but cannot exclude it at present). Finally, we
discuss the situation for arbitrary finite groups which admit an action on a
homology 3-sphere.
